Spivak's Calculus prologue pg 10

After introducing the properties of numbers, Spivak states:

"Note, in particular, that $$a>0$$ if and only if $$a$$ is in $$P$$"

I'm not exactly sure how to prove this. The relevant properties are:

"(P10) (Trichotomy law) For every number $$a$$, one and only one of the following holds:
(i) $$a = 0$$
(ii) $$a$$ is in the collection $$P$$
(iii) $$-a$$ is in the collection $$P$$"

and the definition:

"$$a>b$$ if $$a-b$$ is in $$P$$"

Using a previously introduced property and the definition:

$$a$$ is in $$P \implies a>0$$

But how do I show:

$$a>0 \implies a$$ is in $$P$$?

• we don't all have the book in front of us. What is P? and how was it defined. It sounds like $a > 0$ if and only if $a \in P$ sounds like a definition? Are is there something like you can add something to it to get zero? Or what. – fleablood Feb 16 at 0:12

The definition of $$a \gt 0$$ tells you that $$a-0 \in P$$, and $$a-0=a$$ so $$a-0 \in P \Rightarrow a \in P$$.

• If A then B, does not imply if B then A? – user1390010 Feb 15 at 23:09
• @user1390010 Definitions are if and only if statements, always. – Pedro Tamaroff Feb 15 at 23:10
• That's a good question, but in the context of a definition, @PedroTamaroff is correct. – Robert Shore Feb 15 at 23:13
• OK thanks, just one more clarification: does "A if B" always mean "A if and only if B" or only in the context of a definition? (since it is not in the usual "if B then A" structure) – user1390010 Feb 15 at 23:17
• It really depends on context, but you can pretty much count on that being the case in the context of a definition (because otherwise the language isn't actually defining the concept). – Robert Shore Feb 15 at 23:29